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Impulses in driving semigroups of nonautonomous dynamical systems: Application to cascade systems
1. | Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, São Carlos - SP, 13566-590, Brazil |
2. | Departamento de Matemática, Centro de Ciências Físicas e Matemáticas, Universidade Federal de Santa Catarina, Florianópolis-SC, 88040-900, Brazil |
3. | Faculdade de Matemática, Universidade Federal de Uberlândia, Uberlândia-MG, 38400-902, Brazil |
4. | Departamento de Estatística, Análise Matemática e Optimización & Instituto de Matemáticas, Universidade de Santiago de Compostela, Santiago de Compostela, Spain |
In this paper we investigate the long time behavior of a nonautonomous dynamical system (cocycle) when its driving semigroup is subjected to impulses. We provide conditions to ensure the existence of global attractors for the associated impulsive skew-product semigroups, uniform attractors for the coupled impulsive cocycle and pullback attractors for the associated evolution processes. Finally, we illustrate the theory with an application to cascade systems.
References:
[1] |
N. U. Ahmed,
Existence of optimal controls for a general class of impulsive systems on Banach spaces, SIAM J. Control Optim., 42 (2003), 669-685.
doi: 10.1137/S0363012901391299. |
[2] |
M. Benchora, J. Henderson and S. Ntouyas, Impulsive Differential Equations and Inclusions, Contemporary Mathematics and Its Applications, 2. Hindawi Publishing Corporation, New York, 2006.
doi: 10.1155/9789775945501. |
[3] |
E. M. Bonotto, M. C. Bortolan, T. Caraballo and R. Collegari,
Attractors for impulsive non-autonomous dynamical systems and their relations, J. Differential Equations, 262 (2017), 3524-3550.
doi: 10.1016/j.jde.2016.11.036. |
[4] |
E. M. Bonotto, M. C. Bortolan, T. Caraballo and R. Collegari,
Impulsive non-autonomous dynamical systems and impulsive cocycle attractors, Math. Methods in the Appl. Sci., 40 (2017), 1095-1113.
doi: 10.1002/mma.4038. |
[5] |
E. M. Bonotto and P. Kalita,
On attractors of generalized semiflows with impulses, The Journal of Geometric Analysis, 30 (2020), 1412-1449.
doi: 10.1007/s12220-019-00143-0. |
[6] |
E. M. Bonotto,
Flows of characteristic $0^+$ in impulsive semidynamical systems, J. Math. Anal. Appl., 332 (2007), 81-96.
doi: 10.1016/j.jmaa.2006.09.076. |
[7] |
E. M. Bonotto, M. C. Bortolan, T. Caraballo and R. Collegari,
Impulsive surfaces on dynamical systems, Acta Mathematica Hungarica, 150 (2016), 209-216.
doi: 10.1007/s10474-016-0631-0. |
[8] |
E. M. Bonotto, M. C. Bortolan, A. N. Carvalho and R. Czaja, Global attractors for impulsive dynamical systems - a precompact approach, J. Differential Equations, (2015), 2602-2625.
doi: 10.1016/j.jde.2015.03.033. |
[9] |
M. C. Bortolan, A. N. Carvalho and J. A. Langa,
Structure of attractors for skew product semiflows, J. Differential Equations, 257 (2014), 490-522.
doi: 10.1016/j.jde.2014.04.008. |
[10] |
M. C. Bortolan and J. M. Uzal,
Pullback attractors to impulsive evolution processes: Applications to differential equations and tube conditions, Discrete Contin. Dyn. Syst., 40 (2020), 2791-2826.
doi: 10.3934/dcds.2020150. |
[11] |
B. Bouchard, N.-M. Dang and C.-A. Lehalle,
Optimal control of trading algorithms: A general impulse control approach, SIAM J. Finan. Math., 2 (2011), 404-438.
doi: 10.1137/090777293. |
[12] |
T. Cardinali and R. Servadei,
Periodic solutions of nonlinear impulsive differential inclusions with constraints, Proc. Am. Math. Soc., 132 (2004), 2339-2349.
doi: 10.1090/S0002-9939-04-07343-5. |
[13] |
A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Applied Mathematical Sciences, 182. Springer, New York, 2013.
doi: 10.1007/978-1-4614-4581-4. |
[14] |
V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society Colloquium Publications, 49. American Mathematical Society, Providence, RI, 2002.
doi: 10.1090/coll/049. |
[15] |
S. Dashkovskiy, O. Kapustyan and I. Romaniuk,
Global attractors of impulsive parabolic inclusions, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1875-1886.
doi: 10.3934/dcdsb.2017111. |
[16] |
M. H. A. Davis, X. Guo and G. Wu,
Impulse control of multidimensional jump diffusions, SIAM J. Control Optim., 48 (2010), 5276-5293.
doi: 10.1137/090780419. |
[17] |
M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-Smooth Dynamical Systems: Theory and Applications, Applied Mathematical Sciences, 163. Springer-Verlag London, Ltd., London, 2008.
doi: 10.1007/978-1-84628-708-4. |
[18] |
J. A. Feroe,
Existence and stability of multiple impulse solutions of a nerve equation, SIAM J. Appl. Math., 42 (1982), 235-246.
doi: 10.1137/0142017. |
[19] |
A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, Mathematics and its Applications, 18. Kluwer Academic Publishers Group, Dordrecht, 1988.
doi: 10.1007/978-94-015-7793-9. |
[20] |
W. M. Haddad and Q. Hui,
Energy dissipating hybrid control for impulsive dynamical systems, Nonlinear Anal., 69 (2008), 3232-3248.
doi: 10.1016/j.na.2005.10.052. |
[21] |
S. K. Kaul,
On impulsive semidynamical systems, J. Math. Anal. Appl., 150 (1990), 120-128.
doi: 10.1016/0022-247X(90)90199-P. |
[22] |
K. Li, C. Ding, F. Wang and J. Hu,
Limit set maps in impulsive semidynamical systems, J. Dyn. Control. Syst., 20 (2014), 47-58.
doi: 10.1007/s10883-013-9204-5. |
[23] |
V. F. Rožko,
A certain class of almost periodic motions in systems with pulses, Differencial' nye Uravnenja, 8 (1972), 2012-2022.
|
[24] |
V. F. Rožko, Ljapunov stability in discontinuous dynamical systems, Differencial'nye Uravnenja, 11 (1975), 1005-1012, 1148. |
[25] |
V. F. Rožko, The almost recurrent and recurrent motions of discontinuous dynamical systems, Differencial'nye Uravnenja, 9 (1973), 1826-1830, 1925. |
[26] |
H. Song and H. Wu,
Pullback attractors of nonautonomous reaction-diffusion equations, J. Math. Anal. Appl., 325 (2007), 1200-1215.
doi: 10.1016/j.jmaa.2006.02.041. |
show all references
References:
[1] |
N. U. Ahmed,
Existence of optimal controls for a general class of impulsive systems on Banach spaces, SIAM J. Control Optim., 42 (2003), 669-685.
doi: 10.1137/S0363012901391299. |
[2] |
M. Benchora, J. Henderson and S. Ntouyas, Impulsive Differential Equations and Inclusions, Contemporary Mathematics and Its Applications, 2. Hindawi Publishing Corporation, New York, 2006.
doi: 10.1155/9789775945501. |
[3] |
E. M. Bonotto, M. C. Bortolan, T. Caraballo and R. Collegari,
Attractors for impulsive non-autonomous dynamical systems and their relations, J. Differential Equations, 262 (2017), 3524-3550.
doi: 10.1016/j.jde.2016.11.036. |
[4] |
E. M. Bonotto, M. C. Bortolan, T. Caraballo and R. Collegari,
Impulsive non-autonomous dynamical systems and impulsive cocycle attractors, Math. Methods in the Appl. Sci., 40 (2017), 1095-1113.
doi: 10.1002/mma.4038. |
[5] |
E. M. Bonotto and P. Kalita,
On attractors of generalized semiflows with impulses, The Journal of Geometric Analysis, 30 (2020), 1412-1449.
doi: 10.1007/s12220-019-00143-0. |
[6] |
E. M. Bonotto,
Flows of characteristic $0^+$ in impulsive semidynamical systems, J. Math. Anal. Appl., 332 (2007), 81-96.
doi: 10.1016/j.jmaa.2006.09.076. |
[7] |
E. M. Bonotto, M. C. Bortolan, T. Caraballo and R. Collegari,
Impulsive surfaces on dynamical systems, Acta Mathematica Hungarica, 150 (2016), 209-216.
doi: 10.1007/s10474-016-0631-0. |
[8] |
E. M. Bonotto, M. C. Bortolan, A. N. Carvalho and R. Czaja, Global attractors for impulsive dynamical systems - a precompact approach, J. Differential Equations, (2015), 2602-2625.
doi: 10.1016/j.jde.2015.03.033. |
[9] |
M. C. Bortolan, A. N. Carvalho and J. A. Langa,
Structure of attractors for skew product semiflows, J. Differential Equations, 257 (2014), 490-522.
doi: 10.1016/j.jde.2014.04.008. |
[10] |
M. C. Bortolan and J. M. Uzal,
Pullback attractors to impulsive evolution processes: Applications to differential equations and tube conditions, Discrete Contin. Dyn. Syst., 40 (2020), 2791-2826.
doi: 10.3934/dcds.2020150. |
[11] |
B. Bouchard, N.-M. Dang and C.-A. Lehalle,
Optimal control of trading algorithms: A general impulse control approach, SIAM J. Finan. Math., 2 (2011), 404-438.
doi: 10.1137/090777293. |
[12] |
T. Cardinali and R. Servadei,
Periodic solutions of nonlinear impulsive differential inclusions with constraints, Proc. Am. Math. Soc., 132 (2004), 2339-2349.
doi: 10.1090/S0002-9939-04-07343-5. |
[13] |
A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Applied Mathematical Sciences, 182. Springer, New York, 2013.
doi: 10.1007/978-1-4614-4581-4. |
[14] |
V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society Colloquium Publications, 49. American Mathematical Society, Providence, RI, 2002.
doi: 10.1090/coll/049. |
[15] |
S. Dashkovskiy, O. Kapustyan and I. Romaniuk,
Global attractors of impulsive parabolic inclusions, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1875-1886.
doi: 10.3934/dcdsb.2017111. |
[16] |
M. H. A. Davis, X. Guo and G. Wu,
Impulse control of multidimensional jump diffusions, SIAM J. Control Optim., 48 (2010), 5276-5293.
doi: 10.1137/090780419. |
[17] |
M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-Smooth Dynamical Systems: Theory and Applications, Applied Mathematical Sciences, 163. Springer-Verlag London, Ltd., London, 2008.
doi: 10.1007/978-1-84628-708-4. |
[18] |
J. A. Feroe,
Existence and stability of multiple impulse solutions of a nerve equation, SIAM J. Appl. Math., 42 (1982), 235-246.
doi: 10.1137/0142017. |
[19] |
A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, Mathematics and its Applications, 18. Kluwer Academic Publishers Group, Dordrecht, 1988.
doi: 10.1007/978-94-015-7793-9. |
[20] |
W. M. Haddad and Q. Hui,
Energy dissipating hybrid control for impulsive dynamical systems, Nonlinear Anal., 69 (2008), 3232-3248.
doi: 10.1016/j.na.2005.10.052. |
[21] |
S. K. Kaul,
On impulsive semidynamical systems, J. Math. Anal. Appl., 150 (1990), 120-128.
doi: 10.1016/0022-247X(90)90199-P. |
[22] |
K. Li, C. Ding, F. Wang and J. Hu,
Limit set maps in impulsive semidynamical systems, J. Dyn. Control. Syst., 20 (2014), 47-58.
doi: 10.1007/s10883-013-9204-5. |
[23] |
V. F. Rožko,
A certain class of almost periodic motions in systems with pulses, Differencial' nye Uravnenja, 8 (1972), 2012-2022.
|
[24] |
V. F. Rožko, Ljapunov stability in discontinuous dynamical systems, Differencial'nye Uravnenja, 11 (1975), 1005-1012, 1148. |
[25] |
V. F. Rožko, The almost recurrent and recurrent motions of discontinuous dynamical systems, Differencial'nye Uravnenja, 9 (1973), 1826-1830, 1925. |
[26] |
H. Song and H. Wu,
Pullback attractors of nonautonomous reaction-diffusion equations, J. Math. Anal. Appl., 325 (2007), 1200-1215.
doi: 10.1016/j.jmaa.2006.02.041. |
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